The present application is a U.S. national application of PCT/IL97/00442, filed Dec. 31, 1997.
The invention relates to the determination of the relative spatial position of two surfaces from values of functions on the two surfaces and especially to the determination of the relative position of the planes of two ultrasound tomography slices.
In many ultrasonic imagers, a hand unit is used to scan an object and acquire data for images of tomograph slices of the object. The unit transmits ultrasound pulses into the object and receives reflections of these pulses from structures in the tomograph slices of the object. An operator holds the hand unit and moves it over that portion of the object that is being imaged. Data for an image of a complete slice of the object is acquired rapidly so that as the operator moves the hand unit over the object being imaged, imaging data for many closely spaced slices is acquired.
In order to construct an accurate three dimensional image of the portion of the object scanned, the location and orientation of the tomograph slices with respect to each other must be accurately known. The location of a tomograph slice with respect to the hand unit, and the locations of points, hereafter referred to as xe2x80x9cdata pointsxe2x80x9d, in the tomograph slice at which data are acquired, are accurately known from the structure of the hand unit. However, the structure of the hand unit does not provide any information as to where the tomograph slices are located with respect to each other. In order to determine the relative locations of the tomograph slices, the location and orientation of the hand unit for each tomograph slice for which data is acquired is determined. This is generally done by monitoring the location and orientation of the hand unit with respect to a reference coordinate system fixed in the room in which the ultrasound equipment is operated. The location and orientation of the reference coordinate system is defined by an array of receivers appropriately positioned in the room. The hand unit is equipped with transmitters that transmit signals to these receivers. The location and orientation of the hand unit can be accurately determined from these signals at any time while the hand unit is scanning the object. This system for locating the hand unit and the tomograph slices for which data is acquired is expensive and cumbersome and precludes the possibility of easily moving the ultrasound equipment from place to place.
A mechanical system for locating the position and orientation of a source of ultrasound energy used to produce a three dimensional image an object is described in U.S. Pat. No. 5,487,388 to M. J. Rello et al. In this patent the source described emits a thin planar fan beam of ultrasound energy. The source is fixed to a mechanical stage which rotates the source about a fixed axis of rotation to accurately position the fan beam at different known angles around the fixed axis.
U.S. Pat. No. 5,503,152 to C. G. Oakley et al describes using two elongated arrays of transducers in fixed relation to each other to produce a three dimensional image of an object in a thick volume slice of the object. The slice of the object imaged is an xe2x80x9coverlapxe2x80x9d volume of the object which is illuminated by ultrasound energy from both arrays. The two arrays receive reflections of ultrasound pulses from the same structures in the overlap volume with different time delays. The time delays are translated into distances from the arrays and the structures are located in the overlap region using triangulation and the distance between the arrays. To produce an image of extended volumes, a transducer assembly is mounted to a mechanical device which moves a transducer assembly over a wide range of controllable positions.
It would be advantageous to have a system for locating the relative positions of ultrasound tomograph slices acquired with a mobile hand source of ultrasound energy that did not depend upon locating the position of the moving source with respect to a fixed reference coordinate system.
It is an object of some aspects of the present invention to provide a relatively simple and inexpensive method for accurately locating the relative positions of tomograph slices at which image data for an ultrasound image of an object are acquired.
In a preferred embodiment of the present invention, the relative locations and orientations of tomograph slices are determined by calculation based on the acquired image data. The method determines the location of the slices without the need to determine the location and orientation of the slices with respect to a fixed reference coordinate system.
For each of a plurality of tomograph slices, values for the image data and values for the first spatial derivative of the image data in the direction perpendicular to the plane of the slice are acquired for a plurality of data points in the slice. For each particular slice, the data points are located in the space of the slice by coordinate values measured relative to a coordinate system specific to the particular slice, which coordinate system is defined and known with respect to the plane of the slice. Preferably, the origin of the coordinate system is located at a point in the plane of the slice, the x and y axes are in the plane of the slice and the z axis is perpendicular to the plane of the slice. Spatial derivatives of the image data at data points in the plane of the slice along the x and y axes are calculated from differences between values of the image data at data points in the slice. The coordinate systems specific to different slices are not the same. They may be displaced one from the other and/or their coordinate axes may be rotated with respect to each other.
Since for each slice, the image data and its first spatial derivatives are known at data points in the slice, first order Taylor series can be expanded about these data points that provide estimates of the image data in volumes of space centered at the data points. As a result, for any given tomograph slice, the image data is known in a volume of space centered about the slice, hereafter referred to as the xe2x80x9cTaylor volumexe2x80x9d. The Taylor volume extends about the plane of the tomograph slice to points located at distances from the plane for which the Taylor series expanded about the data points provide substantially accurate estimates of the image data.
In accordance with a preferred embodiment of the present invention, adjacent tomograph slices are located sufficiently close together so that at least one of the adjacent slices, hereafter referred to as the xe2x80x9ctarget slicexe2x80x9d is located inside the Taylor volume of the other slice, hereafter referred to as the xe2x80x9cTaylor slicexe2x80x9d. For a point on the target slice, a measured value of the image data, and a value of the image data for the same point, calculated from a Taylor series expanded about a data point in the Taylor slice, must be the same. In accordance with a preferred embodiment of the present invention, the relative location and orientation of the two adjacent tomograph slices are determined from the requirement that Taylor series expansions of the image data from points in the Taylor slice must provide values for the image data, at points on the target slice, that are substantially the same as the measured values of the image data at the points on the target slice.
In accordance with a preferred embodiment of the present invention, the relative locations and orientations of the specific coordinate systems associated with a plurality of adjacent tomograph slices are determined. Once this is accomplished, image data acquired as functions of the coordinates of the various specific coordinate systems a preferably transformed to a single common coordinate system. The images of the object from the various tomograph slices are then be correlated with each other and an accurate image of the volume of the object containing the tomograph slices is constructed.
Let x,y,z, (in vector notation {right arrow over (x)}), represent the coordinates of points in a Taylor slice with respect to a coordinate system, hereafter referred to as a xe2x80x9cTaylorxe2x80x9d coordinate system, specific to the Taylor slice. Let xxe2x80x2,yxe2x80x2,zxe2x80x2 (in vector notation {right arrow over (x)}xe2x80x2), represent the coordinates of points in a target slice, adjacent to the Taylor slice, with respect to a coordinate system, hereafter a xe2x80x9ctargetxe2x80x9d coordinate system, specific to the target slice. The location and orientation of the target coordinate system with respect to the Taylor coordinate system can be defined by the components, in the Taylor coordinate system, of the vector extending from the origin of the Taylor coordinate system to the origin of the target coordinate system, and angles of rotation of the target coordinate system about the zxe2x80x2, xxe2x80x2, and zxe2x80x2 axes respectively of the target coordinate system. A transform, xe2x80x9cTxe2x80x9d, which is a function of these vector components and angles of rotations, transforms the coordinates of a point in space defined relative to the Taylor coordinate system to the coordinates of the same point defined relative to the target coordinate system. If these vector components and angles of rotation are denoted respectively by, X,Y,Z, xcfx86,xcex8,xcex1, then, {right arrow over (x)}xe2x80x2=T(X,Y,Z,xcfx86,xcex8,xcex1,){right arrow over (x)}=T{right arrow over (x)} and {right arrow over (x)}=Txe2x88x921(X,Y,Z,xcfx86,xcex8,xcex1){right arrow over (x)}xe2x80x2=Txe2x88x921{right arrow over (x)}xe2x80x2.
Let the image data, as functions of the coordinates of a point defined with respect to the Taylor and target coordinate systems, be represented respectively by the values of ultrasound imaging functions F(x,y,z)=F({right arrow over (x)}) and f(xxe2x80x2,yxe2x80x2,zxe2x80x2)=f({right arrow over (x)}xe2x80x2). Then, for a point xe2x80x9cnxe2x80x9d, with coordinates {right arrow over (x)}xe2x80x2|n, in the target coordinate system and {right arrow over (x)}|n=Txe2x88x921{right arrow over (x)}xe2x80x2|n in the Taylor coordinate system f({right arrow over (x)}xe2x80x2)|n=F(Txe2x88x921{right arrow over (x)}xe2x80x2)|n, where the symbol xe2x80x9c|nxe2x80x9d indicates that the expression or quantity is to be evaluated for the point n. If f({right arrow over (x)}xe2x80x2) is measured for at least 6 points n, n=1 . . . 6, and the form of F({right arrow over (x)}) as a function of {right arrow over (x)} is known, then the six equations [f({right arrow over (x)}xe2x80x2)xe2x88x92F(Txe2x88x921{right arrow over (x)}xe2x80x2)]|n=0 for n=1 . . . 6 determine the 6 independent variables, X,Y,Z,xcfx86,xcex8,xcex1, on which T depends and which define the relative location and orientation of the Taylor and target coordinate systems.
In accordance with a preferred embodiment of the present invention, the form of F({right arrow over (x)}xe2x80x2) is known in the Taylor volume of the Taylor slice from its first order Taylor expansions about points on the plane of the Taylor volume. Additionally, in accordance with a preferred embodiment of the present invention, the points on the plane of the target slice for which f({right arrow over (x)}xe2x80x2) is evaluated are located in the Taylor volume of the Taylor slice. As a result, the values of X,Y,Z,xcfx86xcex8,xcex1, which define the relative location and orientation of the two tomograph slices can be calculated.
Generally, the values for the coordinates {right arrow over (x)} and {right arrow over (x)}xe2x80x2, and for measured values of F({right arrow over (x)}xe2x80x2) and f({right arrow over (x)}xe2x80x2) and their derivatives normal to their respective tomograph slice planes, hereafter referred to as xe2x80x9cnormal derivativesxe2x80x9d, are not known exactly but are associated with a degree of uncertainty. For example, the uncertainty with which the functions F({right arrow over (x)}) and f({right arrow over (x)}xe2x80x2) and their respective normal derivatives are known, is much higher for data points further from the source of ultrasound waves used in an ultrasound imaging of an object than for data points closer to the source of ultrasound waves. In a preferred embodiment in accordance with the present invention the values for X,Y,Z,xcfx86,xcex8,xcex1, are calculated from equations which incorporate these uncertainties. Preferably, a set of equations are used to calculate values for X,Y,Z,xcfx86,xcex8,xcex1, which minimize the differences between f({right arrow over (x)}xe2x80x2) and F(Txe2x88x921{right arrow over (x)}xe2x80x2) for a plurality of points n. Preferably, the equations result from a least squares minimization procedure that minimizes the sum   S  =            ∑              n        =        1            N        ⁢          xe2x80x83        ⁢                  (                                            [                                                f                  ⁡                                      (                                                                  x                        →                                            xe2x80x2                                        )                                                  -                                  F                  ⁡                                      (                                                                  T                                                  -                          1                                                                    ⁢                                              x                        →                                                              )                                                              ]                        /                                          σ                ⁡                                  (                                                                                    x                        →                                            xe2x80x2                                        ,                                                                  T                                                  -                          1                                                                    ⁢                                                                        x                          →                                                xe2x80x2                                                                              )                                            2                                "RightBracketingBar"                )            n      
as a function of the variables X,Y,Z,xcfx86,xcex8,xcex1. In the expression for S, N is the number of the plurality of points used in the sum and "sgr"({right arrow over (x)}xe2x80x2,Txe2x88x921{right arrow over (x)}xe2x80x2)|n represents a quantity which is a function of the uncertainties with which {right arrow over (x)}, {right arrow over (x)}xe2x80x2, F({right arrow over (x)}) and f({right arrow over (x)}xe2x80x2) are known for the point n.
Let the variables X,Y,Z,xcfx86,xcex8,xcex1, be represented respectively by the components of a xe2x80x9c6xe2x80x9d vector vk, where k assumes the integer values from 1-6. Then the least squares minimization of S expressed using vk requires that             ∂              {        S        }              /          ∂              v        k              =                    ∂                  {                                                    ∑                                  n                  =                  1                                N                            ⁢                              xe2x80x83                            ⁢                                                (                                                            [                                                                        f                          ⁡                                                      (                                                                                          x                                →                                                            xe2x80x2                                                        )                                                                          -                                                  F                          ⁡                                                      (                                                                                          T                                                                  -                                  1                                                                                            ⁢                                                              x                                →                                                                                      )                                                                                              ]                                        /                                          σ                      ⁡                                              (                                                                                                            x                              →                                                        xe2x80x2                                                    ,                                                                                    T                                                              -                                1                                                                                      ⁢                                                                                          x                                →                                                            xe2x80x2                                                                                                      )                                                                              )                                2                                      ⁢                          |              n                                }                    /              ∂                  v          k                      =    0  
for k=1 . . . 6. These equations can be mathematically quite involved and computationally expensive to solve. In some preferred embodiments of the present invention simplifying assumptions are made which make the least squares procedure for minimizing S more tractable. Preferably, it is assumed that the derivatives of "sgr"({right arrow over (x)}xe2x80x2,Txe2x88x921{right arrow over (x)}xe2x80x2)|n with respect to vk are 0 so that "sgr"({right arrow over (x)}xe2x80x2,Txe2x88x921{right arrow over (x)}xe2x80x2)|n="sgr"({right arrow over (x)}xe2x80x2) (i.e. no dependence upon T), and             ∂              {        S        }              /          ∂              v        k              =                    ∂                  {                                                    ∑                                  n                  =                  1                                N                            ⁢                              xe2x80x83                            ⁢                              (                                                                            [                                                                        f                          ⁡                                                      (                                                                                          x                                →                                                            xe2x80x2                                                        )                                                                          -                                                  F                          ⁡                                                      (                                                                                          T                                                                  -                                  1                                                                                            ⁢                                                                                                x                                  →                                                                xe2x80x2                                                                                      )                                                                                              ]                                        2                                    /                                                            σ                      ⁡                                              (                                                                              x                            →                                                    xe2x80x2                                                )                                                              2                                                  )                                      ⁢                          |              n                                }                    /              ∂                  v          k                      =    0.  
In an alternate preferred embodiment of the present invention, variations in the uncertainties with which {right arrow over (x)}, {right arrow over (x)}xe2x80x2, F({right arrow over (x)}) and f({right arrow over (x)}xe2x80x2) are known as a function of the positions of points n are ignored altogether, and "sgr"({right arrow over (x)}xe2x80x2,Txe2x88x921{right arrow over (x)}xe2x80x2)|n is set equal to 1. The following discussion is independent of whether "sgr" is utilized.
Preferably, N is the smallest integer possible greater than 6, that provides the desired accuracy for the values of vk (i.e. for X,Y,Z,xcfx86,xcex8,xcex1). Preferably, ∂{S}/∂vk=0 is first solved for N equal to a small integer greater than or equal to 6 and then solved in an iterative procedure for increasing values of N until a desired accuracy for vk is reached. In some preferred embodiments in accordance with the present invention trigonometric functions which appear in the transformation T are approximated using small angle approximations of the trigonometric functions.
It is an object of another aspect of the present invention to provide a device for ultrasound imaging of an object that acquires ultrasound image data and values for the normal derivative of the image data at data points in tomograph slices of the object.
An ultrasound imaging device, in accordance with a preferred embodiment of the present invention, comprises a scanning unit for acquiring image data, wherein the scanning unit comprises two parallel linear phased arrays of ultrasound transducers which transmit and receive sound waves used for imaging the object. The two phased arrays are preferably closely spaced one to the other. As a result, when the unit scans the object, values for the image data are acquired for data points on two parallel, very closely spaced, planes through the object. The average of the values of the image data at a first point on one of the planes and at a second point on the other plane adjacent to the first point, provides a value for the image data at the point midway between the first and second points, for the tomograph slice defined by the volume between the two planes. The difference between the values of the image data at the first and second points, divided by the distance between the two planes, provides the derivative of the image data normal to the plane of the tomograph slice at the point midway between the first and second points.
There is therefore provided in accordance with a preferred embodiment of the present invention a method of ultrasound imaging comprising: (a) acquiring ultrasound image data in a first slice defined by a first plane in a first coordinate system; (b) acquiring ultrasound image data in a second slice defined by a second plane in a second coordinate system different from the first coordinate system; (c) determining ultrasound image data in a region adjacent to at least one of said first and second planes; and (e) determining a coordinate transformation between the first and second coordinate systems based on said acquired ultrasound image data and said determined ultrasound image data.
Preferably, determining ultrasound image data in regions adjacent at least one of said first and second planes comprises acquiring image data outside of the plane of the slice. Alternatively or additionally, the method preferably comprises determining spatial derivatives of said image data at points in at least one of said first and second planes, said derivatives comprising derivatives of image data with respect to the normal to the plane. Preferably, said derivatives comprise derivatives of said image data along directions parallel to the plane.
In some preferred embodiments of the present invention determining a coordinate transform comprises: defining a first function of coordinates of the first coordinate system representative of said ultrasound image data of said first slice; defining a second function of coordinates of the second coordinate system representative of said ultrasound image data of said second slice; and requiring that said first and second functions be substantially equal at each of a plurality of points in space. Preferably, defining said first function comprises determining spatial derivatives for said fist slice and defining a plurality of Taylor series wherein each Taylor series is expanded about a point in the plane of the first slice. Preferably, said Taylor series comprises a first order Taylor series.
In some preferred embodiments of the present invention image data at a second point on the plane of the second slice is matched to image data from the first slice and wherein said matching is performed by matching said second slice image data to a Taylor series of image data expanded about a first point on the plane of said first slice. Preferably, said first point has the same coordinates in said first coordinate system as said second point in said second coordinate system.
Preferably, said transformation is derived by matching a plurality of said first points to a plurality of said second points. Preferably, said transformation is derived using an optimization procedure. Additionally or alternatively, points of the plurality of points preferably have a large first derivative. Additionally or alternatively, at least some points of the plurality of points are preferably spatially relatively far apart.
Additionally or alternatively, the number of sets of first and second points is preferably greater than the number of independent parameters required for defining said transformation. Additionally or alternatively, the optimizing procedure is preferably a least squares procedure. Additionally or alternatively, said optimizing procedure incorporates weighting factors that are functions of the accuracy with which locations of and image data values at the first and second points are known. Preferably, the weighting factors are independent of the transformation.
In some preferred embodiments of the present invention, determining said transformation comprises determining said transformation at each iteration of an iterative procedure wherein results of a determination of said transformation in one iteration are used in a determination of the transformation in a subsequent iteration. Preferably, determining said transformation comprises defining at least one measure of the accuracy of the determination of said transformation at each iteration and determining a cutoff value for each of said at least one measure of accuracy. Preferably, said measure of accuracy is a deviations mean square test. Preferably, said measure of accuracy is an F test.
In some preferred embodiments of the present invention, said transformation is applied to data in one of said coordinate systems and including: determining an improved coordinate transformation between the first and second coordinate systems based on said acquired ultrasound image data and said determined ultrasound image data. Preferably, determining an improved coordinate transformation comprises matching image data at a second point on the second slice to image data from the first slice and wherein said matching is performed by matching said second slice image data to a Taylor series of image data expanded about a first point on said first slice. Preferably, said first point has the same coordinates in said first coordinate system as said second point in said second coordinate system after applying said transformation. Additionally or alternatively, said improved coordinate transformation is derived by matching a plurality of said first points to a plurality of said second points in an optimization procedure.
There is further provided in accordance with a preferred embodiment of the present invention an apparatus for three dimensional ultrasound imaging of an object comprising: transducers arrayed in a geometry to acquire ultrasound imaging data in an imaging slice of the object, which imaging slice is defined by a plane, so that data of the acquired data is acquired at points on both sides of the plane of the slice; a processor which processes said data to provide spatial derivatives of said data along directions parallel and perpendicular to the plane of the imaging slice; and wherein said processor determines the relative positions of a plurality of imaging slices using said acquired data and said derivatives to provide a three dimensional image of said object.